Number System (Integers) — Class 7 Maths (Samacheer Kalvi)
TN State Board (Samacheer Kalvi) Class 7 Mathematics, Term 1 — Chapter 1. Working with integers and their properties.
1. About this chapter
This chapter covers integers — their representation on the number line, ordering, the four fundamental operations (addition, subtraction, multiplication, division), and the properties of these operations.
2. Integers and the number line
- Integers are the set … −3, −2, −1, 0, 1, 2, 3 … (positive numbers, negative numbers and zero), written ℤ.
- On the number line, numbers increase to the right and decrease to the left, so for example −5 < −2 < 0 < 3.
3. Operations and the rules of signs
| Operation | Rule | Example |
|---|---|---|
| Add same signs | add, keep the sign | (−4) + (−3) = −7 |
| Add different signs | subtract, take sign of larger | (−7) + 4 = −3 |
| Multiply / divide same signs | answer is positive | (−6) × (−2) = 12 |
| Multiply / divide different signs | answer is negative | (−6) × 2 = −12 |
- A product of an even number of negative integers is positive; of an odd number of negative integers is negative.
4. Properties of operations
For integers a, b, c:
- Closure: a + b, a − b and a × b are always integers (division is not closed).
- Commutative: a + b = b + a and a × b = b × a (subtraction and division are not commutative).
- Associative: (a + b) + c = a + (b + c); (a × b) × c = a × (b × c).
- Distributive: a × (b + c) = a × b + a × c.
- Identity: 0 is the additive identity (a + 0 = a); 1 is the multiplicative identity (a × 1 = a).
5. Worked examples
Example 1. Evaluate (−15) + 8. Different signs → 15 − 8 = 7, take the sign of the larger (15) → −7.
Example 2. Evaluate (−12) ÷ (−4). Same signs → +3.
Example 3. Use the distributive property: 6 × (10 + (−3)). = 6 × 10 + 6 × (−3) = 60 − 18 = 42.
6. Exercises (Samacheer Kalvi)
- Represent −4, 0 and 3 on the number line and order them.
- Find: (a) (−9) + (−6) (b) 15 + (−20) (c) (−8) − (−5).
- Find: (a) (−7) × 4 (b) (−5) × (−9) (c) (−36) ÷ 6 (d) (−42) ÷ (−7).
- Name the property used: (−3) × (4 + 5) = (−3) × 4 + (−3) × 5.
- The temperature was 5 °C and fell by 8 °C. Find the new temperature.
7. Common mistakes
- Mistake: Saying two negatives added give a positive. Fix: (−4) + (−3) = −7 — adding two negatives gives a negative; the product of two negatives is positive.
- Mistake: Thinking integers are closed under division. Fix: (−7) ÷ 2 is not an integer — division is not closed.
- Mistake: Treating subtraction as commutative. Fix: 5 − 3 ≠ 3 − 5.
8. Quick revision
- Term 1 · Ch 1 · integers.
- ℤ = {… −2, −1, 0, 1, 2 …}; on the number line, right = greater.
- Same signs (×, ÷) → positive; different signs → negative; even number of negatives → positive.
- Properties: closure (not ÷), commutative & associative (+, ×), distributive, identities 0 and 1.
